GNOME Shell Integration (menus)
Basically No

No, it's not possible and never will be (in all likelihood).

Blender is basically just one OpenGL Canvas, that uses OpenGL to draw all of its UI using it's own library called GHOST, see Is the Blender GUI package available for other programs?.

Maybe it will be more clear by comparing Blender to a video game using OpenGL (which is essentially the same thing, a pure OpenGL application):
You are asking to put GNOME's menus into Counter Strike/WoW/you name it.

It just won't work because OpenGL is not operating with the normal windowing system of your OS.
Other 3D software like Maya / 3ds max use the native windowing system and only use DirectX/OpenGL to render the 3D Views. Blender as a whole is just one (or more) big OpenGL window(s).

Alternatives

Create additional GNOME windows via python

You could use python to create additional windows using GNOME's native menus, which could then interact with Blender. But that would just be a bad idea, because you'd get into a lot of cross-platform trouble, plus it's inconsistent with the rest of Blender and I don't even know a single project that does something like that.

Use Blender as a python module

If you just want use some functionality of Blender (maybe to modify a lot of blend files in a specific way) and want a GNOME UI then you could use Blender as a python module. That's a bit more involved though. For more info have a look at the Blender Wiki page on using Blender as a python module